Transversal Hamilton cycles in digraph collections

Given a collection $\mathcal{D}=\{D_1,D_2,\dots ,D_m\}$ of digraphs on the common vertex set $V$, an $m$-edge digraph $H$ with vertices in $V$ is $\mathit{\text{transversal}}$ in $\mathcal{D}$ if there exists a bijection $\phi :E(H)\to [m]$ such that $e\in E(D_{\phi (e)})$ for all $e\in E(H)$. Ghouila-Houri proved that any $n$-vertex digraph with minimum semi-degree at least $\frac{n}{2}$ contains a directed Hamilton cycle. In this paper, we provide a transversal generalization of Ghouila-Houri’s theorem, thereby solving a problem proposed by Chakraborti, Kim, Lee and Seo. Our proof utilizes the absorption method for transversals, the regularity method for digraph collections, as well as the transversal blow-up lemma and the related machinery. As an application, when $n$ is sufficiently large, our result implies the transversal version of Dirac’s theorem, which was proved by Joos and Kim.